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Chapter 4: Problem 25

(a) find the vertex and axis of symmetry of each quadratic function. (b) Determine whether the graph is concave up or concave down. (c) Graph the quadratic function. f(x)=-2(x-3)^{2}+5

Short Answer

Expert verified
Vertex: (3, 5). Axis of Symmetry: x = 3. Concave down.

Step by step solution

01

Identify the Standard Form

The quadratic function given is in the vertex form: \[ f(x) = a(x-h)^2 + k \] Here, \( a = -2 \), \( h = 3 \), and \( k = 5 \).
02

Find the Vertex

In the vertex form of a quadratic function, the vertex is the point \( (h, k) \). Substitute the values of \( h \) and \( k \): \[ \text{Vertex} = (3, 5) \]
03

Determine the Axis of Symmetry

The axis of symmetry for a quadratic function in vertex form is the vertical line \( x = h \). Here, \( h = 3 \). So, the equation of the axis of symmetry is: \[ \text{Axis of Symmetry} = x = 3 \]
04

Determine Concavity

The concavity of the graph of a quadratic function is determined by the sign of the coefficient \( a \). If \( a \) is positive, the graph is concave up. If \( a \) is negative, the graph is concave down. Here, \( a = -2 \), which is negative. Thus, the graph is concave down.
05

Plot the Graph

To plot the graph of the function: 1. Plot the vertex at \( (3, 5) \). 2. Draw the axis of symmetry as a vertical line at \( x = 3 \). 3. Since the graph is concave down, draw a parabola opening downward with the vertex at \( (3, 5) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

vertex form
The vertex form of a quadratic function is an incredibly powerful way to understand the properties of the function instantly. The general vertex form is written as: f(x) = a(x-h)^2 + k This format highlights several important components:
  • **a**: This coefficient affects the width and direction of the parabola. If **a** is positive, the parabola opens upwards. If **a** is negative, it opens downwards.
  • **h**: This value affects the horizontal position of the vertex. It鈥檚 the x-coordinate of the vertex.
  • **k**: This is the vertical position of the vertex, representing the y-coordinate.
In the given exercise, the function is: f(x) = -2(x-3)^2 + 5 We can see **a = -2**, **h = 3**, and **k = 5**. This means the vertex of the function is at the point (3, 5) and the parabola opens downwards because **a** is negative.
axis of symmetry
The axis of symmetry for a quadratic function is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. This line helps in graphing the parabola correctly and understanding its structure. To find the axis of symmetry from the vertex form: x = h For our function, since **h = 3**, the axis of symmetry is the line: x = 3 This means the parabola is symmetrical about this vertical line. If you fold the parabola along this line, both halves would align perfectly.
concavity
Concavity describes the direction in which the parabola opens. To determine this, look at the coefficient **a** in the vertex form of the quadratic function.
  • If **a** > 0, the parabola is concave up, resembling a U-shape.
  • If **a** < 0, the parabola is concave down, resembling an upside-down U.
In our example function, **a = -2**. Since **a** is negative, the parabola is concave down. This means it opens downward and the vertex at (3, 5) is the highest point on the graph.
graphing parabolas
Graphing a quadratic function involves a few simple steps that ensure accuracy.
  • **Step 1**: Plot the vertex, which is given by the point (h, k). In our function, the vertex is (3, 5), so plot this point first.
  • **Step 2**: Draw the axis of symmetry as a vertical line through the vertex. For our function, this line is x = 3.
  • **Step 3**: Determine the direction of the parabola鈥檚 opening using the sign of **a**. If **a** is negative, it opens downward. For our example, since **a = -2**, draw a parabola that opens downward, starting from the vertex.
  • **Step 4**: Plot additional points, if needed, to help shape the parabola. Use symmetric points around the vertex on either side of the axis of symmetry to ensure accuracy.
When graphing, always check the consistency of your points and plot them carefully around the axis of symmetry to get the perfect parabolic shape.

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